Grade 2 · 2.MD.D.9 · 3.MD.B.4

Making a dot plot

Ten pencils have been measured, and their lengths sit in a list. How do you turn that list into a picture where every pencil can be seen, sorted and counted?

You need

Reading dot plots: clusters, gaps and peaksGrade 2 Data tables, tallies and frequency tablesGrade 1

Unlocks

Line plots with fractionsGrade 5

Where you are 3 in Dot plots

Statistics7

Data displays12

Dot plots3

Making a dot plot

Reading dot plots: clusters, gaps and peaks Line plots with fractions

When you reach for it

Mereana takes ten pencils from the class pencil cup and measures each one to the nearest inch (Measuring length). She writes the lengths down as she goes: 5, 7, 4, 6, 5, 3, 7, 5, 6, 5. That list is true, but it is hard to see anything in it. Which length came up most? Are the pencils mostly long or mostly short? To answer, you would have to sort and count the list in your head.

A line plot answers those questions with a picture. It is a number line, like the edge of a ruler, with one X above it for each thing measured, placed above that thing’s length. Equal lengths stack up. The same picture is also called a dot plot; this lesson says line plot, as Reading dot plots: clusters, gaps and peaks does. That lesson reads a finished plot. This one draws a plot from a list, which is the same idea run the other way: each number in the list becomes one X.

Reach for a line plot when several things of one kind have been measured with the same unit, or when one thing has been measured several times, and you want to see all the measurements together. The lesson takes three days. Part 1 draws a plot from a list: the number line, the Xs, the title and the label. Part 2 starts earlier, with measuring or with a table, and looks closely at where the number line should start. Part 3 checks a plot against its list, adds new measurements, and reads the finished plot back.

The procedure

Part 1: from a list to a line plot

What goes on the line. Look through the list for the smallest number and the largest. In Mereana’s list the shortest pencil is 3 inches and the longest is 7 inches. Every pencil sits somewhere from 3 to 7, so the number line needs a mark at 3, at 7, and at every whole number in between: 3, 4, 5, 6, 7. A mark goes in for every number in that stretch, even one that no pencil has. The reason is in the next paragraph.

Equal steps. The marks must be the same distance apart, the way inch marks on a ruler are. Each step on the line stands for one more inch (Length on a number line), so the distance between two Xs shows how different the two lengths are. If one step were wider than the rest, two pencils would look further apart than they are. A ruler, the lines of lined paper turned sideways, or the squares of grid paper all make equal steps easy. Write the number under each mark.

Where the line starts. A ruler starts at 0 because it is for measuring, and every length begins at its end. A line plot does not measure anything; the measuring is already done. Its job is to show where the measurements landed, so it only needs the stretch from the smallest to the largest. Marks below the smallest number would stay empty forever, because no measurement is there. Starting at 0 is not wrong, but when the numbers are large it wastes most of the line (Example 4). An extra empty mark at either end is fine too, and leaves room if a new measurement turns up.

Title and label. Under the numbers, write what was measured and the unit: “pencil length (inches)”. Without it, the 5 could mean 5 inches, 5 centimeters or 5 of anything. Over the plot, write a title that says whose things they are or where they came from: “Mereana’s pencils”.

One X for each number. Now go through the list in order, one number at a time. For each number, draw an X above its mark, and cross that number off the list. When a mark already has an X, the new X goes on top of it. Crossing off is what keeps the plot honest: every number is used once, none is skipped and none is drawn twice.

Stack evenly. Make every X the same size and leave the same space between Xs in every stack. Then the height of a stack shows how many Xs it holds, and two stacks can be compared at a glance, just like two bars in a bar graph. Squares of grid paper help again: one X per square.

Count to check. When the list is all crossed off, count the Xs. There must be exactly as many Xs as numbers in the list, because each number made one X. Mereana’s list has 10 numbers, so her plot must have 10 Xs.

Some plots use dots instead of Xs. Either one works, as long as each thing measured gets exactly one.

Part 2: where the list comes from, and choosing the line

Measuring first. The list is made by measuring. Lay each object along a ruler or a yardstick with one end at 0, see which whole-unit mark the other end is nearest, and write that number down (Measuring length). Every object must be measured in the same unit, all in inches or all in centimeters, because one number line has one unit. A list can come from many objects, such as the pencils in a cup, or from one object measured again and again, such as a table that every child in the class measures in turn. The plot is drawn the same way in both cases.

From a table instead of a list. Sometimes the measurements have already been counted into a table, with each length beside the number of things that had it (Data tables, tallies and frequency tables). A table saves the crossing off: the count beside each length says how many Xs to stack above its mark. A table often leaves out a length that nothing had. The number line still needs that mark, for the same reason as before: the steps must stay equal.

Choosing the line, again. The smallest and largest numbers decide the line, whatever size the numbers are. If a table was measured nine times and every answer was between 46 and 50 inches, the line runs from 46 to 50: five marks. A line from 0 would need 51 marks, and all nine Xs would be squeezed into the last five.

Part 3: checking, adding, and reading back

Checking stack by stack. Counting all the Xs catches a missing X or an extra one, but not an X on the wrong mark: one X moved from the 6 to the 5 leaves the total the same. So check each mark too. Find how many times its number appears in the list, and count the Xs above it. The two must agree at every mark. A mark whose number is not in the list must have no Xs.

Adding a measurement. A new measurement is one new X, stacked on its mark, and nothing else moves. If the new number is past either end of the line, the line grows: add marks, at the same step, until the new number has one.

Reading it back. A finished plot is read the way Reading dot plots: clusters, gaps and peaks teaches: the tallest stack is the most common length, the Xs farthest left and right are the shortest and longest, and the Xs to the right of a mark are the things longer than it. A question or two at the end shows the point of all the drawing: the answer is now a matter of looking, not of sorting the list again.

Worked examples

Example 1

Part 1: Mereana’s pencils

Mereana measured 10 pencils to the nearest inch: 5, 7, 4, 6, 5, 3, 7, 5, 6, 5. Make a line plot.

Solution

The smallest number in the list is 3 and the largest is 7, so the number line runs from 3 to 7 with a mark at every whole number: 3, 4, 5, 6, 7. The marks are spaced equally, one inch per step. Under the line goes “pencil length (inches)”, and on top the title.

Figure 1The number line comes first. It runs from the shortest pencil to the longest, one inch per step.

Now take the list one number at a time. The first number is 5: draw an X above the 5 and cross off the 5. Next is 7: an X above the 7. Then 4, then 6. The fifth number is 5 again, so its X goes on top of the X already above the 5. Carry on to the end of the list.

Figure 2One X for each of Mereana’s 10 pencils, stacked above its length.

The stacks are 1 above 3, 1 above 4, 4 above 5, 2 above 6 and 2 above 7. Counting the Xs,

1 + 1 + 4 + 2 + 2 = 10

which matches the 10 numbers in the list. The tallest stack is above the 5, so 5 inches is the most common pencil length, something the list never showed at a glance.

Example 2

Part 1: a mark with nothing above it

Niko picks 12 snap pea pods and measures each one to the nearest centimeter: 8, 6, 9, 7, 8, 10, 6, 8, 9, 8, 7, 12. Make a line plot.

Solution

The shortest pod is 6 centimeters and the longest is 12, so the line runs from 6 to 12: 6, 7, 8, 9, 10, 11, 12. No pod in the list is 11 centimeters long, but 11 is between 6 and 12, so it gets a mark. Leaving it out would put the 12 right next to the 10, and a pod 2 centimeters longer would look only one step longer.

Label the line “pod length (centimeters)” and give the plot a title. Then go through the list, drawing one X for each number and crossing it off.

Figure 3No pod is 11 centimeters long, but the mark at 11 stays.

The stacks are 2, 2, 4, 2, 1 at 6 to 10, nothing at 11, and 1 at 12:

2 + 2 + 4 + 2 + 1 + 0 + 1 = 12

Twelve Xs for twelve pods. The empty mark is not a mistake; it says that no pod was 11 centimeters long.

Example 3

Part 2: from a table

Petelo picked snap pea pods too, and counted his into a table. Make a line plot.

Pod length (centimeters)	Number of pods
5	2
6	4
7	3
9	1

Solution

The table already sorted and counted the pods, so there is no list to cross off. The shortest length in the table is 5 and the longest is 9, so the line runs from 5 to 9. The table has no row for 8, because no pod had that length, but the line needs a mark at 8 to keep its steps equal.

Each row says how many Xs to stack: 2 above the 5, 4 above the 6, 3 above the 7, and 1 above the 9.

Figure 4Drawn from a table. The table has no row for 8, but the line still has a mark there.

The table’s counts add to

2 + 4 + 3 + 1 = 10

and the plot has 10 Xs, one for each pod. The table and the plot hold the same facts. The plot adds what the table hides: the empty step at 8, which puts the 9 a little apart from the rest.

Example 4

Part 2: one table, measured nine times

Nine children each measure the length of the art table with a tape measure, to the nearest inch. Their answers are 47, 48, 47, 46, 47, 50, 47, 48, 47. Make a line plot. Should the line start at 0? About how long is the table?

Solution

The smallest answer is 46 and the largest is 50, so the line runs from 46 to 50: five marks, 46, 47, 48, 49, 50. The line could start at 0, and that would not be wrong, but a line from 0 to 50 would need 51 marks, and the first 46 of them would never get an X. Every answer is near 47, so the useful part of the line is the short stretch around it. Starting at 46 keeps the marks wide enough to stack on and the picture easy to read.

Going through the list gives 1 X above 46, 5 above 47, 2 above 48, none above 49 and 1 above 50.

Figure 5Nine children measured the same table. The line starts at 46, not at 0.

Check: 1 + 5 + 2 + 0 + 1 = 9, one X for each child.

The children measured one table, so the answers differ only because measuring by hand is never perfect. Five of the nine said 47 inches, and the rest are close by, so the table is about 47 inches long. The X at 50 stands a step apart from the others. That measurement is worth doing again: perhaps the tape slipped.

Example 5

Part 3: does the plot match the list?

Niko picks a second batch of 10 pods: 7, 9, 8, 7, 10, 9, 7, 8, 9, 6 centimeters. A plot of them is drawn below. Does it match the list?

Figure 6Does this plot match the list? Check it stack by stack.

Solution

First count all the Xs: 1 + 3 + 2 + 2 + 1 = 9. The list has 10 numbers, so the plot is missing one X. To find which, check mark by mark.

Mark	Times in the list	Xs on the plot
6	1	1
7	3	3
8	2	2
9	3	2
10	1	1

Every mark agrees except 9: the list has three 9s and the plot has two Xs. One more X above the 9 fixes the plot. Then it has 1 + 3 + 2 + 3 + 1 = 10 Xs, and every mark matches.

Crossing off each number as its X is drawn prevents exactly this. A 9 that was skipped stays uncrossed on the list, where it is easy to spot.

Example 6

Part 3: two more pods

Niko finds two more pods from his first picking, 5 centimeters and 10 centimeters long. Add them to his first plot (Example 2).

Solution

Each new pod is one new X, and the old Xs stay where they are. The 10-centimeter pod has a mark already, so its X goes on top of the one X above the 10, making a stack of 2.

The 5-centimeter pod is shorter than any pod so far, and the line starts at 6. So the line grows one step to the left, a new mark at 5 spaced exactly like the others, and the X goes above it. If the line was drawn on paper with no room on the left, it has to be drawn again; leaving an empty mark at each end the first time saves that.

Figure 7Two new pods. The line grows one step to the left to make room for the 5.

The plot now holds

12 + 2 = 14

Xs. The shortest pod is now 5 centimeters, the longest is still 12, and the mark at 11 is still empty.

Example 7

Part 3: from start to finish, then two questions

Petelo measures 11 pencils to the nearest centimeter: 12, 15, 17, 15, 14, 18, 15, 12, 17, 19, 15. Make a line plot. How many pencils are 15 centimeters or longer? How much longer is the longest pencil than the shortest?

Solution

The smallest number is 12 and the largest is 19, so the line has a mark at every whole number from 12 to 19. That includes 13 and 16, which are not in the list. Label it “pencil length (centimeters)”, title it, and draw one X for each number, crossing off as you go.

Figure 8Petelo’s 11 pencils, from the list to the finished plot.

The stacks are 2 above 12, 1 above 14, 4 above 15, 2 above 17, 1 above 18 and 1 above 19. Check: 2 + 1 + 4 + 2 + 1 + 1 = 11, the length of the list.

Now read it. The pencils 15 centimeters or longer are the stack at 15 and every X to its right:

4 + 2 + 1 + 1 = 8

So 8 pencils are 15 centimeters or longer. The longest pencil is the X farthest right, 19, and the shortest is the X farthest left, 12. Counting the steps from 12 to 19, or subtracting,

19 - 12 = 7

so the longest pencil is 7 centimeters longer than the shortest.

Boundaries

Numbers only. A line plot is built on a number line, so it needs measurements that are numbers in order. Favorite colors or kinds of fruit have no order and no distance between them; they go in a picture graph or a bar graph.
One unit for the whole plot. Pencils measured in inches and pencils measured in centimeters cannot share a line, because a 5-inch pencil and a 5-centimeter pencil are not the same length. Measure everything in one unit, or make two plots.
Whole units here. Every measurement in this lesson is rounded to the nearest whole inch or centimeter, so the marks are whole numbers one unit apart. Two pencils in the same stack may differ by a little; the plot is only as exact as the measuring.
Every step between the ends. The line may start at the smallest measurement and stop at the largest, but between them no whole number may be skipped, or the equal steps break.
One X for one measurement. The count of Xs must equal the count of numbers in the list. That check is needed but not enough: an X on the wrong mark keeps the total right, so a full check goes mark by mark.
The plot keeps lengths, not names. Once drawn, the plot does not say which pencil had which length, or the order they were measured in.

Where this goes next

The finished plot is read with the tools of Reading dot plots: clusters, gaps and peaks: stacks, the most common length, longest and shortest, clusters, gaps and peaks. Its neighbor, Making bar graphs, draws counts of named groups; a stack of Xs is a bar made of separate pieces.

Later the marks get finer. Measuring to the half inch and quarter inch (Measuring to the half and quarter inch) gives a number line marked in halves and quarters, and Line plots with fractions draws and uses those plots. In middle school the same drawing is used for any list of numbers at all, test scores, travel times or family sizes, and the shape of the stacks becomes the shape of a distribution.

Exercises

Part 1: from a list to a line plot

Mereana measures 7 more pencils, in inches: 4, 6, 5, 6, 7, 6, 4. What is the smallest length? The largest? Which numbers go under the marks of the number line?
Make a line plot of the list in Exercise 1. How many Xs are above each mark?
Niko measures 7 pods, in centimeters: 11, 8, 9, 11, 10, 11, 9. Make a line plot.
Make a line plot of these pencil lengths, in inches: 3, 5, 5, 6, 3, 5, 8, 6. Which marks have no Xs above them?
Write a title and a label for the plot in Exercise 4.
A list has 13 measurements. How many Xs will its line plot have?

Part 2: measuring, tables, and choosing the line

Make a line plot from this table of pencil lengths.

Pencil length (centimeters)	Number of pencils
10	1
11	3
12	2
14	2

Make a line plot from this table of snap pea pods. How many pods are there?

Pod length (centimeters)	Number of pods
4	1
5	3
6	5
7	2
9	1

Eight children measure the width of the reading rug, to the nearest inch: 59, 60, 59, 59, 58, 59, 60, 59. Make a line plot. About how wide is the rug?
A pencil’s end lies between the 5-inch and 6-inch marks of a ruler, closer to the 6. What number goes on the list?
Measurements of a rug run from 57 to 61 inches. How many marks does the line need if it runs from the smallest to the largest? How many if it starts at 0?
A number line for a plot has the marks 3, 4, 5, 7, 8, equally spaced. The list has no 6 in it. Is the line right?

Part 3: checking, adding, and reading back

Use this plot for Exercise 13.

Figure 9A plot to check, for Exercise 13.

This list of pencil lengths, in inches, was plotted in the figure above: 4, 5, 5, 6, 4, 7, 5, 6, 5. Does the plot match the list? If not, how should it change?
Two more pods, 7 and 11 centimeters long, join Niko’s pods from Exercise 3. How does the plot change? How many Xs does it have now?
Mereana finds one more pencil, 9 inches long, and adds it to her plot from Exercise 2. What changes on the number line? How many Xs are there now?
Make a line plot of these pod lengths, in centimeters: 9, 6, 7, 9, 10, 9, 6, 9, 8. What is the most common length?
Use your plot from Exercise 16. How many pods are longer than 7 centimeters? How much longer is the longest pod than the shortest?
Make a line plot of these pencil lengths, in centimeters: 16, 13, 18, 13, 17, 13, 18. Which marks are empty? How much longer is the longest pencil than the shortest?

Explain

Why does a mark stay on the line even when no measurement has that number?
Why may a line plot’s number line start at the smallest measurement instead of at 0?
Mereana measured her pencils in inches and Petelo measured his in centimeters. Why can’t their pencils go on one line plot?
A plot has exactly as many Xs as its list has numbers. Does that prove the plot is right?
Why should every X be the same size, with the same space between Xs in every stack?

Answers

Smallest 4 inches, largest 7 inches. The marks are 4, 5, 6, 7, every whole number from the smallest to the largest.
A line from 4 to 7 labeled “pencil length (inches)”. Xs: 2 above 4, 1 above 5, 3 above 6, 1 above 7. Check: 2 + 1 + 3 + 1 = 7, one X per pencil.
A line from 8 to 11 labeled “pod length (centimeters)”. Xs: 1 above 8, 2 above 9, 1 above 10, 3 above 11. Check: 1 + 2 + 1 + 3 = 7.
A line from 3 to 8. Xs: 2 above 3, 3 above 5, 2 above 6, 1 above 8. The marks at 4 and 7 have no Xs, but they stay on the line. Check: 2 + 3 + 2 + 1 = 8.
Any title saying whose pencils they are, such as “Our pencils”, and the label “pencil length (inches)” under the numbers. The label must name the unit.
13 Xs. Each measurement makes exactly one X.
A line from 10 to 14, with a mark at 13 even though the table skips it. Xs: 1 above 10, 3 above 11, 2 above 12, none above 13, 2 above 14. Check: 1 + 3 + 2 + 2 = 8.
A line from 4 to 9, with an empty mark at 8. Xs: 1, 3, 5, 2 above 4 to 7 and 1 above 9. There are 1 + 3 + 5 + 2 + 1 = 12 pods.
A line from 58 to 60. Xs: 1 above 58, 5 above 59, 2 above 60; 1 + 5 + 2 = 8. The rug is about 59 inches wide: most answers say 59 and the rest are one inch away.
6. The pencil’s end is nearest the 6-inch mark, so to the nearest inch it is 6 inches long.
5 marks, 57, 58, 59, 60, 61, since 61 - 57 + 1 = 5. From 0 it needs 62 marks, 0 to 61, and the first 57 would all be empty.
No. The step from 5 to 7 is two inches but is drawn the same size as a one-inch step. A mark at 6 must go in, even with no X above it.
No. Both have 9 Xs, but the list has four 5s and two 6s, while the plot has 3 Xs above 5 and 3 above 6. Move one X from the 6 stack to the 5 stack. The stacks should be 2, 4, 2, 1 above 4, 5, 6, 7.
The line grows one step left, with a new mark at 7 and one X above it. The stack at 11 grows from 3 to 4. The plot now has 7 + 2 = 9 Xs.
The line must reach 9, so marks at 8 and 9 are added at the same step. The 8 stays empty and one X goes above 9. There are now 7 + 1 = 8 Xs.
A line from 6 to 10. Xs: 2 above 6, 1 above 7, 1 above 8, 4 above 9, 1 above 10; 2 + 1 + 1 + 4 + 1 = 9. The most common length is 9 centimeters, the tallest stack.
6 pods are longer than 7 centimeters: the Xs right of the 7 are 1 + 4 + 1 = 6. The longest is 10 and the shortest 6, and 10 - 6 = 4 centimeters.
A line from 13 to 18. Xs: 3 above 13, 1 above 16, 1 above 17, 2 above 18; 3 + 1 + 1 + 2 = 7. The marks at 14 and 15 are empty. The longest is 18 - 13 = 5 centimeters longer than the shortest.
The marks must be equally spaced, one unit per step, so that the distance between two Xs shows how different their lengths are. Without the empty mark, two Xs on either side would look one step apart when they are two. The empty mark also says something true: nothing had that length.
The plot does not measure; it shows where measurements landed. No measurement is below the smallest one, so marks below it would always be empty. Starting at the smallest keeps the line short and the stacks easy to see.
A 5-inch pencil is much longer than a 5-centimeter pencil, so the two cannot share the mark 5. One line has one unit. Either measure all the pencils in the same unit or make two plots.
No. An X on the wrong mark keeps the total right, as in Exercise 13. The count of Xs must match, and so must every stack, mark by mark.
Then the height of a stack shows how many Xs are in it. If the Xs in one stack were bigger or more spread out, a stack of 3 could look taller than a stack of 4.